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Theorem r19.3rm 3511
Description: Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.)
Hypothesis
Ref Expression
r19.3rm.1 𝑥𝜑
Assertion
Ref Expression
r19.3rm (∃𝑦 𝑦𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem r19.3rm
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2240 . . 3 (𝑎 = 𝑦 → (𝑎𝐴𝑦𝐴))
21cbvexv 1918 . 2 (∃𝑎 𝑎𝐴 ↔ ∃𝑦 𝑦𝐴)
3 eleq1 2240 . . . 4 (𝑎 = 𝑥 → (𝑎𝐴𝑥𝐴))
43cbvexv 1918 . . 3 (∃𝑎 𝑎𝐴 ↔ ∃𝑥 𝑥𝐴)
5 biimt 241 . . . 4 (∃𝑥 𝑥𝐴 → (𝜑 ↔ (∃𝑥 𝑥𝐴𝜑)))
6 df-ral 2460 . . . . 5 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
7 r19.3rm.1 . . . . . 6 𝑥𝜑
8719.23 1678 . . . . 5 (∀𝑥(𝑥𝐴𝜑) ↔ (∃𝑥 𝑥𝐴𝜑))
96, 8bitri 184 . . . 4 (∀𝑥𝐴 𝜑 ↔ (∃𝑥 𝑥𝐴𝜑))
105, 9bitr4di 198 . . 3 (∃𝑥 𝑥𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))
114, 10sylbi 121 . 2 (∃𝑎 𝑎𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))
122, 11sylbir 135 1 (∃𝑦 𝑦𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1351  wnf 1460  wex 1492  wcel 2148  wral 2455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-cleq 2170  df-clel 2173  df-ral 2460
This theorem is referenced by:  r19.28m  3512  r19.3rmv  3513  r19.27m  3518  indstr  9579
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